I've fortunately never been in a situation like this family, though I've been involved in decisions to discontinue life support measures for very elderly relatives.
However, I'm uncomfortably reminded of a difficult few months when my youngest was a baby. When she was 6 weeks old we found out she probably had a rare condition that would need surgery by a neurosurgeon. I was petrified. My sister has profound and multiple learning difficulties and I was so scared my baby would too. Her appointment wouldn't be for another 3 months and I just wasn't coping. One day I drove my baby in tears to the specialist unit and demanded to be seen. They didn't move her appointment forward but I was treated with such compassion, they took the time to talk to me and explain the delay wouldn't cause any issues. My baby was taken by the play specialist for a cuddle so she wouldn't hear me cry and brought back for hugs when I was calmer. In retrospect, I was putting my needs before my baby's. All she cared about was that I fed her, cuddled her, interacted with her. She was 'healthy'. But I was consumed with this fear.
A few months later she had her appointment, then the surgery and, as scary as it was, it all went well and she's now a healthy 14yo. But I'll never forget the compassion everyone showed towards a terrified mum who really wasn't coping, probably wasn't putting her baby first (even though she was fed, clothed, clean, hugged a bit too much if anything) and was taking up their valuable time unnecessarily. The HV was also great, she delayed doing any PND tests till after surgery as she said they wouldn't be accurate as I was under stress.
My point from all this is that I am sure the staff are used to dealing with parents in distress and are doing the best they can. I also recognise that at times of stress even good parents act irrationally. For Archie's sake, I hope this comes to an end soon as it doesn't seem possible he is going to recover and the whole family need to heal.